Article
KYUNGPOOK Math. J. 2019; 59(4): 725-734
Published online December 23, 2019
Copyright © Kyungpook Mathematical Journal.
Integral Formulas Involving Product of Srivastava’s Polynomials and Galu´e type Struve Functions
Daya Lal Suthar∗ and Mitku Andualem
Department of Mathematics, Wollo University, P.O. Box: 1145, Dessie, Ethiopia
e-mail : dlsuthar@gmail.com and andualemmitku@gmail.com
Received: March 5, 2018; Revised: August 1, 2019; Accepted: August 5, 2019
Abstract
The aim of this paper is to establish two general finite integral formulas involving the product of Galué type Struve functions and Srivastava’s polynomials. The results are given in terms of generalized (Wright’s) hypergeometric functions. These results are obtained with the help of finite integrals due to Oberhettinger and Lavoie-Trottier. Some interesting special cases of the main results are also considered. The results presented here are of general character and easily reducible to new and known integral formulae.
Keywords: Galué, type Struve function, gamma function, generalized Wright function, Oberhettinger integral formula, Lavoie-Trottier integral formula.
1. Introduction and Preliminaries
Nisar et al., in [10], defined Galué type Struve functions (GTSF) as a generalization of Struve functions as follows:
where
If we set
Details related to the function
Recall that the generalized Wright hypergeometric function
The generalized Wright function was introduced by Wright [24] in the form of (
It is noted that the generalized (Wright) hypergeometric function
where
where (
ℤ0 denotes the set of nonpositive integers.
Now, we recall the following known functions. Srivastava’s polynomials are defined is [18] as
where
For our present investigation, we also need to recall the following Oberhettinger integral formula [12]:
provided 0 < ℜ(
Also we recall the Lavoie-Trottier integral formula from [7]:
provided ℜ(
2. Main Results
The main purpose of this paper is to introduce four generalized integral formulas involving products of general class of polynomials and generalized Galué type Struve functions. The integral formulas are as follows:
Theorem 2.1
Using (
We can apply the integral formula (
In accordance with the definition of (
Theorem 2.2
Proceeding as in the proof of Theorem 2.1, we get the integral formula (
Theorem 2.3
Using (
We can apply the integral formula (
In accordance with the definition of (
Theorem 2.4
Proceeding as in the proof of Theorem 2.3, we get the integral formula (
3. Special Cases
In this section, we derive in Corollaries 3.1 – 3.4 some new integral formulae by using known generalized Struve function. We also derive as example a result involving the Hermite polynomials.
If we employ the same method as in proofs of Theorems 2.1 – 2.4, we obtain the following four corollaries with the help of (
Corollary 3.1
Corollary 3.2
Corollary 3.3
Corollary 3.4
Further, If we set
where
In this case for example, Theorems 2.1 and 2.3 yield the following results involving the Hermite polynomials and the generalized Struve functions.
Corollary 3.5
Corollary 3.6
Remark 3.7
If we set
4. Conclusion
In the present paper, we used generalized (Wright) hypergeometric functions to investigated new integrals involving the generalized Struve functions and Srivastava polynomials. Certain special cases of involving Struve functions have been investigated in the literature by a number of authors [21, 22, 23] using different arguments. The results presented in this paper can easily be altered to deal with similar new interesting integrals by making suitable parameter substitutions. Further, for given suitable special values for the coefficient
Acknowledgements
The authors are thankful to the referee for his/her valuable remarks and comments for the improvement of the paper.
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